From Riemann Manifolds to Riemann Manifolds - Map Projections: Cartographic Information Systems - page 96

canonical measures are exclusively used to generate in following sections equidistant, conformal,
and equiareal mappings of various surfaces like the ellipsoid-of-revolution to the sphere. Hilbert's
invariant theory is finally used to generate scalar functions of the tensor-valued deformation
measures. Box 1.47 reviews the two fundamental Hilbert invariants of the Cauchy-Green arid
Euler-Lagrange deformation tensors.

2
l

2

Box 1.46 (Canonical criteria for a conformal, equiareal, and isometric mapping M → M

r as

well as for an equidistant mapping c l ( t ) → c r ( t )).

Conformeomorphism:
Λ 1 = Λ 2 or λ 1 = λ 2 ,

(1.300)

K 1 = K 2 or κ 1 = κ 2 ,
for all points of

r , respectively .
Areomorphism :
Λ 1 Λ 2 =1 or λ 1 λ 2 =1 ,

l or

M M

(1.301)

K 1 K 2 + 1

2 ( K 1 + K 2 )=0 or κ 1 κ 2 + 1 2 ( κ 1 + κ 2 )=0 ,

2 2

for all points of

r , respectively .
Isometry :
Λ 1 = Λ 2 =1 or λ 1 = λ 2 =1 ,

M

l or

M

(1.302)

K 1 = K 2 =0 or κ 1 = κ 2 =0 ,
for all points of

1 , respectively .
Equidistance:
Λ 1 =1 , Λ 2 =1or λ 1 =1 , λ 2 =1 .

l or

M M

(1.303)

K 1 =0 , K 2 =0or κ 1 =0 , 2 =0 .

for all points of M

l (left curve) and M r (right curve) which are equidistantly mapped.

Box 1.47 (Canonical representation of Hilbert invariants derived from deformation measures).

I 1 (C l ):= Λ 1 + Λ 2 =tr[C l G − l ]

s i 1 (C r ):= λ 1 + λ 2 =tr[C r G − r ] ,

(1.304)

I 2 (C l ):= Λ 1 Λ 2 =det[C l G − l ] i 2 (C r ):= λ 1 λ 2 =det[C r G − r ] ,

ers s

or

I 1 (E l ):= K 1 + K 2 = tr[E l G − l ] s i 1 (E r ):= κ 1 + κ 2 = tr[E r G − r ] ,

(1.305)

i 2 (E r ):= κ 1 κ 2 =det[E r G − r ] . I 2 (E l ):= K 1 K 2 =det[E l G l 1] versus